Generalized Curvatures: 2 (Geometry and Computing) by Jean-Marie Morvan

By Jean-Marie Morvan

The vital item of this ebook is the degree of geometric amounts describing N a subset of the Euclidean house (E ,), endowed with its commonplace scalar product. allow us to nation accurately what we suggest by means of a geometrical volume. think about a subset N S of issues of the N-dimensional Euclidean house E , endowed with its common N scalar product. LetG be the gang of inflexible motions of E . we are saying zero volume Q(S) linked toS is geometric with recognize toG if the corresponding zero volume Q[g(S)] linked to g(S) equals Q(S), for all g?G . for example, the zero diameter ofS and the world of the convex hull ofS are amounts geometric with admire toG . however the distance from the foundation O to the nearest aspect ofS isn't, zero because it isn't really invariant lower than translations ofS. you will need to indicate that the valuables of being geometric relies on the selected team. for example, ifG is the 1 N team of projective adjustments of E , then the valuables ofS being a circle is geometric forG yet now not forG , whereas the valuables of being a conic or a directly zero 1 line is geometric for bothG andG . This perspective can be generalized to any zero 1 subsetS of any vector area E endowed with a groupG performing on it.

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